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Mirrors > Home > MPE Home > Th. List > cardeq0 | Structured version Visualization version GIF version |
Description: Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.) |
Ref | Expression |
---|---|
cardeq0 | ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5203 | . . 3 ⊢ ∅ ∈ V | |
2 | carden 9962 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → ((card‘𝐴) = (card‘∅) ↔ 𝐴 ≈ ∅)) | |
3 | 1, 2 | mpan2 687 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = (card‘∅) ↔ 𝐴 ≈ ∅)) |
4 | card0 9376 | . . 3 ⊢ (card‘∅) = ∅ | |
5 | 4 | eqeq2i 2834 | . 2 ⊢ ((card‘𝐴) = (card‘∅) ↔ (card‘𝐴) = ∅) |
6 | en0 8561 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
7 | 3, 5, 6 | 3bitr3g 314 | 1 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3495 ∅c0 4290 class class class wbr 5058 ‘cfv 6349 ≈ cen 8495 cardccrd 9353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-ac2 9874 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-wrecs 7938 df-recs 7999 df-er 8279 df-en 8499 df-card 9357 df-ac 9531 |
This theorem is referenced by: tskcard 10192 |
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